Multi-element Generalized Polynomial Chaos Research Articles

Robust optimization of nonlinear energy sinks used for mitigation of friction-induced limit cycle oscillations

This paper aims at proposing robust methods to optimize nonlinear energy sinks (NES) used for the mitigation of friction-induced vibrations due to mode coupling instabilities. The study is based on a mechanical system composed of two NES coupled to the well-known two-degrees-of-freedom Hultèn’s model. In such an unstable system coupled with NES, it is usual to observe a discontinuity in the steady-state amplitude profile which separates the parameter space into two parts corresponding respectively to the mitigated and unmitigated regimes. The discontinuity is predicted by a methodology previously developed by the authors and based on Multi-Element generalized Polynomial Chaos. The method allows to determine the Propensity of the system to undergo a Harmless Steady-State Regime (PHSSR). The objective of the present work is therefore to maximize the value of the PHSSR to obtain a robust optimal design of the NES. To this end, several stochastic optimization problems are presented that take into account the dispersion of the uncertain parameters using two approaches; in the first one, the parameters of the NES are considered as deterministic, and in the second one they are also supposed uncertain but with a known probability law.

Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

Abstract We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.

A polynomial chaos method for arbitrary random inputs using B-splines

Isogeometric analysis which extends the finite element method through the usage of B-splines has become well established in engineering analysis and design procedures. In this paper, this concept is considered in context with the methodology of polynomial chaos as applied to computational stochastic mechanics. In this regard it is noted that many random processes used in several applications can be approximated by the chaos representation by truncating the associated series expansion. Ordinarily, the basis of these series are orthogonal Hermite polynomials which are replaced by B-spline basis functions. Further, the convergence of the B-spline chaos is presented and substantiated by numerical results. Furthermore, it is pointed out, that the B-spline expansion is a generalization of the Legendre multi-element generalized polynomial chaos expansion, which is proven by solving several stochastic differential equations.

Prediction of the dynamic behavior of an uncertain friction system coupled to nonlinear energy sinks using a multi-element generalized polynomial chaos approach

In this paper, a friction system with uncertain parameters and coupled to two Nonlinear Energy Sinks (NESs) is studied. The dispersion of some physical parameters due to their uncertain nature may generate a dynamic instability which leads to a Limit Cycle Oscillations (LCO) causing a propensity of squeal. The concept of Targeted Energy Transfer (TET) by means of NESs to mitigate this squealing noise is proposed. In this kind of unstable dynamical system coupled to NES, the transition from harmless regimes (i.e. the LCO is mitigated) to harmful regimes (i.e. the LCO is not mitigated) as a function of the uncertain parameters implies a discontinuity in the steady-state amplitude profiles. In this context, a Multi-Element generalized Polynomial Chaos (ME-gPC) based method is proposed to locate this discontinuity (called mitigation limit) and therefore to predict the Propensity of the system to undergo an Harmless Steady-State Regime (PHSSR). The results obtained with this original method lead to a good compromise between computational cost and accuracy in comparison with a reference method.

Level set methods for stochastic discontinuity detection in nonlinear problems

Stochastic problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zeros of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.

Pitfalls in the frequency response represented onto Polynomial Chaos for random SDOF mechanical systems

• The frequency response function of a stochastic single-degree-offreedom oscillator is studied. • Conditions for which the probability density function is multimodal are established. • It is shown that Polynomial Chaos then fails to provide accurate results. • Outputbased approach and multi-element generalized Polynomial Chaos are shown to return satisfying results. Uncertainties are present in the modeling of dynamical systems and they must be taken into account to improve the prediction of the models. It is very important to understand how they propagate and how random systems behave. This study aims at pointing out the somehow complex behavior of the structural response of stochastic dynamical systems and consequently the difficulty to represent this behavior using spectral approaches. The main objective is to find numerically the probability density function (PDF) of the response of a random linear mechanical systems. Since it is found that difficulties can occur even for a single-degree-of-freedom system when only the stiffness is random, this work focuses on this application to test several methods. Polynomial Chaos performance is first investigated for the propagation of uncertainties in several situations of stiffness variances for a damped single-degree-of-freedom system. For some specific conditions of damping and stiffness variances, it is found that numerical difficulties occur for the standard polynomial bases near the resonant frequency, where it is generally observed that the shape of the system response PDFs presents multimodality. Strategies to build enhanced bases are then proposed and investigated with varying degrees of success. Finally, a multi-element approach is used in order to gain robustness.

Stability analysis of a clutch system with multi-element generalized polynomial chaos

In the transmission systems of vehicles, unforced vibrations can be observed during the sliding phase of clutch engagement. These vibrations are due to frictional forces and may generate noise. Several studies have shown that the stability of such friction systems is highly sensitive to parameters (friction law, damping) which lead to significant dispersions. Therefore, uncertain parameters must be considered in the stability analysis of a clutch system. This paper investigates the ability of the multi-element generalized polynomial chaos to take an increasing number of uncertain parameters into account in the stability analysis of a clutch system: it focuses on accuracy, on the criterion for the choice of the order of truncation, on the criterion for the choice of the number of elements and on computation time. The objective is to propose a low-cost model of high accuracy, compared with the Monte Carlo method.

Wiener–Askey and Wiener–Haar Expansions for the Analysis and Prediction of Limit Cycle Oscillations in Uncertain Nonlinear Dynamic Friction Systems

This paper is devoted to the robust modeling and prediction of limit cycle oscillations in nonlinear dynamic friction systems with a random friction coefficient. In recent studies, the Wiener–Askey and Wiener–Haar expansions have been proposed to deal with these problems with great efficiency. In these studies, the random dispersion of the friction coefficient is always considered within intervals near the Hopf bifurcation point. However, it is well known that friction induced vibrations—with respect to the distance of the friction dispersion interval to the Hopf bifurcation point—have different properties in terms of tansient, frequency and amplitudes. So, the main objective of this study is to analyze the capabilities of the Wiener–Askey (general polynomial chaos, multielement generalized polynomial chaos) and Wiener–Haar expansions to be efficient in the modeling and prediction of limit cycle oscillations independently of the location of the instability zone with respect to the Hopf bifurcation point.

Multi-element stochastic spectral projection for high quantile estimation

We investigate quantile estimation by multi-element generalized Polynomial Chaos (gPC) metamodel where the exact numerical model is approximated by complementary metamodels in overlapping domains that mimic the model’s exact response. The gPC metamodel is constructed by the non-intrusive stochastic spectral projection approach and function evaluation on the gPC metamodel can be considered as essentially free. Thus, large number of Monte Carlo samples from the metamodel can be used to estimate α-quantile, for moderate values of α. As the gPC metamodel is an expansion about the means of the inputs, its accuracy may worsen away from these mean values where the extreme events may occur. By increasing the approximation accuracy of the metamodel, we may eventually improve accuracy of quantile estimation but it is very expensive. A multi-element approach is therefore proposed by combining a global metamodel in the standard normal space with supplementary local metamodels constructed in bounded domains about the design points corresponding to the extreme events. To improve the accuracy and to minimize the sampling cost, sparse-tensor and anisotropic-tensor quadratures are tested in addition to the full-tensor Gauss quadrature in the construction of local metamodels; different bounds of the gPC expansion are also examined. The global and local metamodels are combined in the multi-element gPC (MEgPC) approach and it is shown that MEgPC can be more accurate than Monte Carlo or importance sampling methods for high quantile estimations for input dimensions roughly below N=8, a limit that is very much case- and α-dependent.

Prediction of Random Self Friction-Induced Vibrations in Uncertain Dry Friction Systems Using a Multi-Element Generalized Polynomial Chaos Approach

The prediction of self friction-induced vibrations is of major importance in the design of dry friction systems. This is known to be a challenging problem since dry friction systems are very complex nonlinear systems. Moreover, it has been shown that the friction coefficients admit dispersions depending in general on the manufacturing process of dry friction systems. As the dynamic behavior of these systems is very sensitive to the friction parameters, it is necessary to predict the friction-induced vibrations by taking into account the dispersion of friction. So, the main problem is to define efficient methods which help to predict friction-induced vibrations by taking into account both nonlinear and random aspect of dry friction systems. The multi-element generalized polynomial chaos formalism is proposed to deal with this question in a more general setting. It is shown that, in the case of friction-induced vibrations obtained from long time integration, the proposed method is efficient by opposite to the generalized polynomial chaos based method and constitutes an interesting alternative to the prohibitive Monte Carlo method.

Probabilistic evaluation of the aerodynamic properties of a bridge deck

The study of the probabilistic response of wind-excited structures and their reliability analysis need to take into account for the influence of a large number of uncertain parameters ascribed to meteorological data, structural characteristics and aerodynamic/aeroelastic behaviour. Although the latter one plays a key role in the reliability analysis and it shows high sensitivity to a number of parameters, the probabilistic description of the aerodynamic and aeroelastic forces remains scarce in literature. The present work aims at giving a contribution to the probabilistic evaluation of the aerodynamic properties of a bluff body, namely a bare bridge deck with trapezoidal cross section. The section lower edge degree of sharpness and the oncoming turbulence integral length scale are retained as random variables. The computational solution of the resulting stochastic Navier–Stokes equations is performed by means of the adaptive form of the Multi-Element generalized Polynomial Chaos. As a result, the stochastic aerodynamic coefficients are described by their approximate probability density functions, which are related in turn to the aerodynamic regimes recognised for the case study.

Stochastic heat transfer enhancement in a grooved channel

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$ , which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

Predicting shock dynamics in the presence of uncertainties

We revisit the classical aerodynamics problem of supersonic flow past a wedge but subject to random inflow fluctuations or random wedge oscillations around its apex. We first obtain analytical solutions for the inviscid flow, and subsequently we perform stochastic simulations treating randomness both as a steady as well as a time-dependent process. We use a multi-element generalized polynomial chaos (ME-gPC) method to solve the two-dimensional stochastic Euler equations. A Galerkin projection is employed in the random space while WENO discretization is used in physical space. A key issue is the characteristic flux decomposition in the stochastic framework for which we propose different approaches. The results we present show that the variance of the location of perturbed shock grows quadratically with the distance from the wedge apex for steady randomness. However, for a time-dependent random process the dependence is quadratic only close to the apex and linear for larger distances. The multi-element version of polynomial chaos seems to be more effective and more efficient in stochastic simulations of supersonic flows compared to the global polynomial chaos method.

Long-term behavior of polynomial chaos in stochastic flow simulations

In this paper we focus on the long-term behavior of generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC) for partial differential equations with stochastic coefficients. First, we consider the one-dimensional advection equation with a uniform random transport velocity and derive error estimates for gPC and ME-gPC discretizations. Subsequently, we extend these results to other random distributions and high-dimensional random inputs with numerical verification using the algebraic convergence rate of ME-gPC. Finally, we apply our results to noisy flow past a stationary circular cylinder. Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior.

Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures

We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.

Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs

In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures. Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each element and subsequently implement generalized Polynomial Chaos (gPC) locally. Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures

An adaptive multi-element generalized polynomial chaos method for stochastic differential equations

We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized polynomial chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the Kraichnan–Orszag three-mode problem, as well as advection–diffusion problems. The new method is similar to spectral element method for deterministic problems but with h– p discretization of the random space.